Properties

Label 2-45e2-1.1-c3-0-109
Degree $2$
Conductor $2025$
Sign $1$
Analytic cond. $119.478$
Root an. cond. $10.9306$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5·2-s + 17·4-s − 9·7-s + 45·8-s − 8·11-s − 43·13-s − 45·14-s + 89·16-s + 122·17-s − 59·19-s − 40·22-s + 213·23-s − 215·26-s − 153·28-s + 224·29-s − 36·31-s + 85·32-s + 610·34-s − 206·37-s − 295·38-s + 413·41-s + 392·43-s − 136·44-s + 1.06e3·46-s + 311·47-s − 262·49-s − 731·52-s + ⋯
L(s)  = 1  + 1.76·2-s + 17/8·4-s − 0.485·7-s + 1.98·8-s − 0.219·11-s − 0.917·13-s − 0.859·14-s + 1.39·16-s + 1.74·17-s − 0.712·19-s − 0.387·22-s + 1.93·23-s − 1.62·26-s − 1.03·28-s + 1.43·29-s − 0.208·31-s + 0.469·32-s + 3.07·34-s − 0.915·37-s − 1.25·38-s + 1.57·41-s + 1.39·43-s − 0.465·44-s + 3.41·46-s + 0.965·47-s − 0.763·49-s − 1.94·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(119.478\)
Root analytic conductor: \(10.9306\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2025,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(7.020645674\)
\(L(\frac12)\) \(\approx\) \(7.020645674\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 5 T + p^{3} T^{2} \)
7 \( 1 + 9 T + p^{3} T^{2} \)
11 \( 1 + 8 T + p^{3} T^{2} \)
13 \( 1 + 43 T + p^{3} T^{2} \)
17 \( 1 - 122 T + p^{3} T^{2} \)
19 \( 1 + 59 T + p^{3} T^{2} \)
23 \( 1 - 213 T + p^{3} T^{2} \)
29 \( 1 - 224 T + p^{3} T^{2} \)
31 \( 1 + 36 T + p^{3} T^{2} \)
37 \( 1 + 206 T + p^{3} T^{2} \)
41 \( 1 - 413 T + p^{3} T^{2} \)
43 \( 1 - 392 T + p^{3} T^{2} \)
47 \( 1 - 311 T + p^{3} T^{2} \)
53 \( 1 - 377 T + p^{3} T^{2} \)
59 \( 1 - 337 T + p^{3} T^{2} \)
61 \( 1 - 40 T + p^{3} T^{2} \)
67 \( 1 + 348 T + p^{3} T^{2} \)
71 \( 1 - 62 T + p^{3} T^{2} \)
73 \( 1 - 1214 T + p^{3} T^{2} \)
79 \( 1 + 294 T + p^{3} T^{2} \)
83 \( 1 + 534 T + p^{3} T^{2} \)
89 \( 1 + 810 T + p^{3} T^{2} \)
97 \( 1 - 928 T + p^{3} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.744442920862777094786493653173, −7.54302440507720158239308747616, −7.05331286361592483273461601475, −6.17525838466062319051760894976, −5.41610477277240314671884484251, −4.82061796892134327160407654394, −3.90102363482918596208993553748, −3.01210032652314390477902613684, −2.45195064075438523820067725167, −0.942282092694781997146685569466, 0.942282092694781997146685569466, 2.45195064075438523820067725167, 3.01210032652314390477902613684, 3.90102363482918596208993553748, 4.82061796892134327160407654394, 5.41610477277240314671884484251, 6.17525838466062319051760894976, 7.05331286361592483273461601475, 7.54302440507720158239308747616, 8.744442920862777094786493653173

Graph of the $Z$-function along the critical line